No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

Jauch–Piron states in *W**‐algebraic quantum mechanics

Rent:

Rent this article for

USD

10.1063/1.527775

### Abstract

A state φ on a *W**‐algebra M is said to fulfill the *J* *a* *u* *c* *h*–*P* *i* *r* *o* *n* *c* *o* *n* *d* *i* *t* *i* *o* *n* if φ(*p*)=φ(*q*)=1 for projections *p*,*q*∈M implies φ(*p*∧*q*)=1. Here *p*∧*q* denotes the infimum of *p* and *q* in the projection lattice of M. The Jauch–Piron condition is a compatibility condition between the algebraic and the lattice‐theoretic approach for the description of physical systems. Normal (i.e., σ‐weakly continuous) states always fulfill the Jauch–Piron condition. It is argued that states not fulfilling this condition should be regarded as unphysical. It is shown that a state φ on a σ‐finite *f* *a* *c* *t* *o* *r*M is singular if and only if projections *e*, *f*∈ M exist such that φ(*e*)=φ( *f* )=1 and *e*∧*f*=0. In particular, any *p* *u* *r* *e* state φ on M fulfilling the Jauch–Piron condition is normal, which implies that the underlying factor M is of type I. Furthermore, the following result is proved: Let φ be a *p* *u* *r* *e* Jauch–Piron state on *W**‐algebra M with separable predual and without any commutative summand. Then φ is normal and a central projection *z* _{0}∈ M exists such that φ(*z* _{0})=1 and *z* _{0} M*z* _{0} is a factor of type I. Thus, *c* *u* *m* *g* *r* *a* *n* *o* *s* *a* *l* *i* *s*, *p* *u* *r* *e* Jauch–Piron states exist only on commutative *W**‐algebras and type I factors. The former case corresponds to classical theories, the latter to Hilbert‐space quantum mechanics. The implications of these results on the interpretation of quantum mechanics are discussed.

© 1987 American Institute of Physics

Received 13 February 1987
Accepted 10 June 1987

/content/aip/journal/jmp/28/10/10.1063/1.527775

http://aip.metastore.ingenta.com/content/aip/journal/jmp/28/10/10.1063/1.527775

Article metrics loading...

/content/aip/journal/jmp/28/10/10.1063/1.527775

1987-10-01

2014-07-24

Full text loading...

### Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content